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  • 8/12/2019 Optimization-Based Safety Analysis of Obstacle Avoidance Systems for UAV

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    J Intell Robot Syst (2012) 65:219231

    DOI 10.1007/s10846-011-9586-0

    Optimization-Based Safety Analysis of Obstacle

    Avoidance Systems for UnmannedAerial Vehicles

    Sivaranjini Srikanthakumar Cunjia Liu

    Wen Hua Chen

    Received: 16 February 2011 / Accepted: 18 April 2011 / Published online: 9 September 2011 Springer Science+Business Media B.V. 2011

    Abstract The integration of Unmanned Aerial

    Vehicles (UAVs) in airspace requires new meth-

    ods to certify collision avoidance systems. This pa-

    per presents a safety clearance process for obsta-

    cle avoidance systems, where worst case analysis is

    performed using simulation based optimization in

    the presence of all possible parameter variations.

    The clearance criterion for the UAV obstacle

    avoidance system is defined as the minimum dis-

    tance from the aircraft to the obstacle during the

    collision avoidance maneuver. Local and globaloptimization based verification processes are de-

    veloped to automatically search the worst com-

    binations of the parameters and the worst-case

    distance between the UAV and an obstacle under

    all possible variations and uncertainties. Based on

    a 6 Degree of Freedom (6DoF) kinematic and

    dynamic model of a UAV, the path planning and

    collision avoidance algorithms are developed in

    S. Srikanthakumar (B) C. Liu W. H. ChenDepartment of Aeronautical and AutomotiveEngineering, Loughborough University,LE11 3TU, Loughborough, UKe-mail: [email protected]

    C. Liue-mail: [email protected]

    W. H. Chene-mail: [email protected]

    3D space. The artificial potential field method is

    chosen as a path planning and obstacle avoidance

    candidate technique for verification study as it is

    a simple and widely used method. Different opti-

    mization algorithms are applied and compared in

    terms of the reliability and efficiency.

    Keywords Clearance process

    Obstacle avoidance Optimization

    Potential field method

    Unmanned aerial vehicle

    1 Introduction

    Due to the absence of a pilot, the use of UAVs

    has become increasingly popular in military and

    civilian applications. Path planning of UAVs with

    known and unknown obstacles is considered as

    one of the key enabling technologies in unmanned

    vehicle systems. Indeed, a significant amount of

    research has been devoted to this subject in re-

    cent years. In addition to offering better per-

    formance, the main industrial concern related to

    new methods is to reduce the risk of collision

    in the presence of all possible parameter varia-

    tions and various failure conditions. Therefore, all

    proposed collision avoidance algorithms have to

    be verified under all operational conditions and

    variations that may be experienced during the life

    of the UAVs. The objective of this paper is to de-

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    220 J Intell Robot Syst (2012) 65:219231

    velop a process to support safety-critical obstacle

    avoidance systems for UAV operation. The cer-

    tification process essentially aims at providing the

    evidence in order to certify that the aircraft is safe

    to fly in the presence of obstacles and parameter

    variations. This task is a very time consuming and

    expensive process, particularly for aircraft [1].Without a pilot, computer algorithms must be

    developed to generate a feasible path in real time.

    Depending on the operation scenarios, there are

    different path planning methods. An UAV has to

    find a collision-free path between the departure

    and the destination (or a waypoint) configurations

    in a static and dynamic environment containing

    various obstacles. Several algorithms have been

    applied to path planning for UAV in the presence

    of known obstacles. Ray tracing and limit cycle

    navigation is combined in [2] for UAV opera-tion in both 2D and 3D space, while Griffiths

    et al. present a rapidly exploring random tree

    (RRT) based path planner through 3D environ-

    ment for an autonomous aerial vehicle [3]. In

    [4], both probabilistic roadmap-based and RRT

    algorithms are used for generating 3D collision

    free path for an autonomous helicopter. Bortoff

    develops a collision free path planning method

    using Voronoi graph search method [5], whereas

    a model predicative control based trajectory op-

    timization method is used to avoid obstacles forNap-of-the-Earth flight in [6]. UAV motion plan-

    ning techniques based on potential field functions

    have been extensively studied; e.g. [7, 8]. UAV

    path planning using the artificial potential field

    method will be used as a candidate collision avoid-

    ance technique in this paper for safety assessment.

    Three major concerns in regard to autonomous

    vehicle operation are efficiency, safety and ac-

    curacy. As the safety of autonomous vehicles is

    dependent on control systems and obstacle avoid-

    ance algorithms, it must be proven that the con-

    trol systems and obstacle avoidance algorithms

    function correctly in the presence of all possible

    vehicle and environmental variations. Two par-

    ticular difficulties faced by designers are a mis-

    matching between the model used for algorithm

    development and the real vehicle dynamics, and

    various uncertainties in vehicle operations. To

    simplify the process of the algorithm develop-

    ment, in general a much simplified model that

    captures the main characters of a vehicle is used in

    the design stage under a number of assumptions

    or simplifications. This causes the mismatching

    between the model used in the design and real

    vehicle behavior. Furthermore, the variations of

    the autonomous vehicle dynamics in operation

    may arise due to the changes of the vehicle itself(e.g. the change of mass or the centre of gravity)

    or the change of the operational environment. As-

    sessment of the safety must be performed not only

    on the nominal model, but also for all possible

    vehicle and environmental variations, and in the

    presence of the mismatching between the model

    used for the design and the real vehicle. There-

    fore, techniques and procedure are demanded to

    understand the behavior of UAVs in the presence

    of such uncertainties. They must cover all possi-

    ble combinations of UAV parameters so that toguarantee that the worst-case performance is ad-

    equate, which is particularly important for safety

    critical functions such as collision avoidance.

    Fault Tree Analysis method was applied to

    the TCAS (Traffic Alert and Collision Avoidance

    Systems) for UAVs safety analysis in [9], while

    Failure Modes and Effects Analysis (FMEA) was

    used [10]. In [11], Functional Failure Analysis

    (FFA) was performed for safety analysis of UAV

    operation including collision avoidance. Two crit-

    ical hazards in UAV operation were defined inthis analysis: midair collision and ground impact.

    Obstacle Analysis was applied to rotorcraft UAV

    in [12], where potential side effects and missing

    monitoring and control requirements were iden-

    tified by step-by-step use of the obstacle analysis

    technique. In [13], the Markov Decision Process

    and Observable Markov Decision Process solvers

    was proposed to generate avoidance strategies

    optimizing a cost function that balances flight-plan

    deviation with anti-collision. The performance of

    this collision avoidance system was evaluated us-

    ing a simulation framework developed for TCAS

    studies. A framework for provably safe decentral-

    ized trajectory planning of multiple aircraft was

    presented in [14]. Each aircraft plans its trajec-

    tory individually using a receding horizon strategy

    based on mixed integer linear programming.

    In this study, the minimum distance from the

    aircraft to an obstacle during a collision avoid-

    ance maneuver is chosen as the criterion for the

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    J Intell Robot Syst (2012) 65:219231 221

    performance assessment. To successfully perform

    collision avoidance maneuvers, the minimum dis-

    tance to the obstacle (dmin) must be greater than

    the radius of the obstacle (rn) including a safe

    margin. The worst case analysis in the presence of

    all the possible uncertainties is cast as a problem

    to find the combinations of the variations wherethe minimum distance to the obstacle (dmin) ap-

    pears. Instead of exhaustive searching the worst

    cases as in many practices, it is well known that

    optimization can effectively find a (local) mini-

    mum or maximum without evaluating all possible

    parameters in a solution space [15]. In this paper,

    the worst case analysis for collision avoidance

    algorithms is treated as a constrained nonlinear

    optimization problem with simulation being in-

    volved in each iteration. In order to pass the safety

    assessment of an anti-collision system,dminin theworst cases must be greater than rn(dmin>rn) in

    the presence of all possible uncertain parameter

    variations. Otherwise, the obstacle avoidance al-

    gorithm and associated controllers have to be re-

    designed to satisfy the anti-collision specification.

    The proposed approach in this paper is applied

    to the collision avoidance algorithm using an ar-

    tificial potential field method for a simple UAV

    model. However, the basic idea is applicable for

    other unmanned vehicles with collision avoidance

    algorithms developed by other methods. It shallbe highlighted that it is not the intention of this

    paper to refine or develop a collision avoidance

    methods. Instead, it is to develop a new procedure

    to support the safety assessment of an existing

    collision avoidance algorithm for UAV operating

    in 3D environment.

    The rest of the paper is organized as follows:

    the simplified kinematic and dynamic model of

    a 6 DOF UAV is introduced in Section 2. The

    clearance criterion of obstacle avoidance is also

    discussed in this section. Motion planning and

    collision avoidance algorithms in 3D environment

    are designed in Section 3using the artificial po-

    tential field method. In Section 4, the obstacle

    avoidance algorithm is validated at nominal pa-

    rameters. In Section5, initial robustness analysis

    of the collision avoidance algorithm is carried out,

    where the optimization based verification process

    is introduced and local optimization algorithms

    are first presented. Two stochastic global opti-

    mization algorithm based verification processes

    are developed in Section6. One is genetic algo-

    rithms (GA) and the other GLOBAL algorithm.

    Furthermore, in order to guarantee finding the

    worst-cases, a deterministic global optimization

    method, i.e. DIviding RECTangles (DIRECT),

    is applied to the worst case analysis of thecollision avoidance algorithm in Section 7. Sim-

    ulation results are presented to verify the pro-

    posed verification processes. Finally, Section 8

    concludes the paper and outlines future research

    directions.

    2 UAV Model and Clearance Criterion

    2.1 UAV Model

    In order to present a clearance criterion of

    obstacle avoidance systems, an UAV model is

    considered for the algorithm development and

    assessment. A six-degree-of-freedom (6DOF)

    rigid-body aircraft model is considered in this

    study. The configuration vector s= (x,y, z, ,

    , ) is used to specify the position and orienta-

    tion of the UAV in the global coordination, where

    q0 =(x,y, z)is the c.g. (center of gravity) position

    of the vehicle and = ( , , ) are the Euler

    angles, with as the roll, as the pitch, and as

    the yaw.

    The kinematic model is given by the following

    two equations [7,8]:

    xy

    z

    =

    c c css s c ss + c csss c c +sss ss c cs

    s cc cc

    v1

    (1)

    =

    1 s t c t0 c s0 s /c c /c

    v2 (2)

    where velocities are described in a body fixed

    frame with liner velocity v1 = [u v w]T and an-

    gular velocity v2 = [p q r]T. Furthermore, the

    following notations are used: s. sin(.), c.

    cos(.), t. tan(.).

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    The longitudinal and lateral dynamics for a

    fixed wing aircraft are given by the linear equa-

    tions[18]

    u = Xuu +Xw W+ Xqq +X+Xe e+ Xt

    w = Zu

    u +Zw W+ Zq

    q +Z+Ze

    e+ Z

    t

    q = Muu +Mw W+ Mqq +M+Me e+ Mt

    v =Yv v+ Ypp (u0 Yr) r+ Y + Y

    +Ya a+ Yrr

    p= Lv v+ Lpp +Lrr+La a+ Lrr

    r= Nvv+ Npp +Nrr+Na a+ Nrr (3)

    where con = [e, a, r, ]T are control in-

    puts, corresponding to the elevator, aileron, rud-

    der deflection angles, and thrust, respectively.The stability and control derivatives used in

    this dynamic model are derived from a nonlin-

    ear UAV model using linearization. Therefore,

    these derivatives depend on the physical parame-

    ters and aerodynamic coefficients of the UAV.

    Several derivatives are of particular interest in

    this study and used as uncertainty parameters,

    which are Xu,Xw,Zu,Zw,Xt,Ze, Yv, Ya, Yrbeing inversely proportional to the aircraft mass

    (m), Xt and Me proportional to aerodynamic

    coefficients ofCtand Cmerespectively.

    2.2 Clearance Criterion

    An UAV has to find a collision-free path between

    the starting pointing and the goal (e.g. waypoint)

    in an environment containing various static ob-

    stacles. Specifically, spherical obstacles are con-

    sidered in this study but it is applicable to other

    obstacles. To assess the safety of UAVs, the min-

    imum distance to the obstacle (dmin) is defined

    as the clearance criterion in the time domain. In

    order to maintain safety, as shown in Fig. 1, the

    influence range of an obstacle is determined from

    the radius of the obstacle plus a specified safe mar-

    gin. The specified safe margin can be chosen ac-

    cording to the UAVs dimensions. As mentioned

    above, the artificial potential field method is used

    in developing obstacle avoidance algorithms in

    Fig. 1 Obstacle avoidance clearance criterion

    this paper. In this framework, obstacle avoidance

    maneuver can be performed within the repulsive

    potential field influence range of an obstacle,

    where the UAV is repulsed from the obstacle and

    attracted to its goal position.

    For a spherical obstacle, the influence range is

    chosen as the radius ofrinflwhich is greater than

    the radius of the obstacle (r0) and the safe mar-

    gin(rsaf e). Letting rn =r0+ rsaf e, the anti-collision

    condition is defined as dmin >rn. In the obsta-

    cle avoidance clearance process, all violations ofthese clearance criteria must be found and the

    worst-case result for each criterion computed. The

    corresponding worst-case combination of uncer-

    tainties must also be computed.

    3 Motion Control and Obstacle Avoidance

    Figure2provides an overview of the motion plan-

    ning and control architecture [16]. The goal of

    motion planning is to generate a desired trajectory

    so that the UAV can track. Aircraft longitudinal

    and lateral dynamics and kinematic equations are

    considered for the clearance process. The high

    level mission planer usually supplies waypoint in-

    formation to the motion controller. Then the mo-

    tion planner retrieves the waypoints and generates

    a desired trajectory. The inner-loop control law

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    Fig. 2 Motion planningand control structure

    qv1,v2, o

    Outer-Loop

    Inner-Loop

    conMotion

    Controller

    Inner-Loop

    Controller Dynamics Kinematics

    (Xd)ud

    d

    is responsible to compute the input signals that

    drive the motors and control surfaces to force

    the UAV to fly at a desired linear velocity and

    attitude so the collision avoidance path generated

    by the motion controller can be followed.

    3.1 Motion Controller Using Potential

    Field Method

    The potential field method was first used by

    Khatib for manipulators and mobile robots path

    planning in the 1980s [19]. The basic concept

    of the potential field method is to fill the ro-

    bots workspace with an artificial potential field

    in which the robot is attracted to its goal position

    and repulsed away from any obstacles. The UAV

    path planning is, in a sense, similar to that of a

    mobile robot. The combination of the attractive

    force to the goal and repulsive forces away from

    any obstacles drives the UAV in a safe path to the

    goal.

    Let q0 =(x,y, z) denote the current UAV

    point in airspace. The usual choice for the attrac-

    tive potential is the standard parabolic that grows

    quadratically with the distance to the goal, such

    that

    Uatt(q0)= 1

    2kad

    2

    goal(q0) (4)

    where dgoal=

    q0 Xgoal

    is the Euclidean dis-tance of the UAVs current position q0 to the

    goal Xgoal and ka is a scaling factor [17,20]. The

    gradient is calculated as

    Uatt(q0)= ka

    q0 Xgoal

    (5)

    The attractive force considered in the potential

    field based approach is the negative gradient of

    the attractive potential

    Fatt(q0)= Uatt(q0)= ka

    q0 Xgoal

    (6)

    By setting the vehicle velocity vector proportional

    to the vector field force, the force Fatt(q0) drives

    the UAV to the goal with a velocity that decreases

    when the UAV approaches the goal.

    The repulsive potential keeps the vehicle away

    from obstacles. This repulsive potential is stronger

    when the UAV is closer to the obstacles and has a

    decreasing influence when the UAV is far away. A

    possible repulsive potential generated by obstacleiis

    Urepi (q0) =

    1

    2krep

    1

    dobsti (q0) 1

    do

    2, if dobsti(q0) d0

    0 , if dobsti(q0) > d0

    (7)

    wherei is the number of the obstacle close to the

    UAV, dobsti (q0) is the closest distance to the ob-

    staclei,kris a scaling constant andd0is the obsta-

    cle influence threshold. The negative gradient of

    the repulsive potential,Frepi(q0)= Urepi(q0), isgiven by,

    Frepi (q0)=

    kr

    1

    dobsti (q0) 1

    d0

    1

    d2obsti(q0)

    ei, if dobsti(q0) d0

    0 , if dobsti (q0)> d0

    (8)

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    where ei = dobsti (q0)

    (q0) is a unit vector that indi-

    cates the direction of the repulsive force [21].

    Therefore,

    xdyd

    zd

    = Uatt(q0) + Urepi (q0)

    =

    Fatt(q0) +Frepi (q0)

    , (9)

    After the desired global velocity is calculated

    by the potential field method, the correspond-

    ing desired linear velocity ud and attitude d =

    (d, d, d)can also be obtained based on UAVs

    kinematic model using the following equations:

    ud =ku

    x2d+ y

    2

    d+ z2

    d

    (10)

    d = 0 (11)

    d =atan2

    zd,

    x2d+ y

    2

    d

    (12)

    d =atan2 ( yd, xd) (13)

    where gain kuis introduced to allow for additional

    freedom in weighting the velocity commands. The

    pitch and yaw angle guidance laws are designed so

    that the vehicles longitudinal axis steers to alignwith the gradient of the potential field. The roll

    angle guidance law is designed to maintain the

    level flight.

    3.2 Inner-Loop Controller

    To accomplish the goal of driving the UAV flying

    at the desired linear velocity ud and desired at-

    titude angles d, the first step is to compute

    the error between the true linear, the attitude

    angles and the desired ones, respectively. Tothis effect, let eu =(ud u), ea =(d ), ee =

    (d )and er=(d )denote the linear ve-

    locity and attitude angle errors, respectively. A

    simple PID control law is proposed as

    =kpe +Ki

    to

    e () d+Kdde

    dt (14)

    Four PID controllers are designed for controlling

    linear velocity and three attitude angles, respec-

    tively. As the angular rates p, q and r are avail-

    able in flight control, the corresponding derivative

    terms in the PID controllers are replaced by their

    angular rate feedback.

    4 Collision Avoidance Validation

    at Nominal Case

    In this section, the proposed collision avoidance

    algorithm and controller are validated at the nom-

    inal parameters. The simulation results for a UAV

    approaching a spherical obstacle are presented at

    the nominal parameters. The nominal parameter

    values are m= 1.9 kg , Cme = 1.13, and Ct=

    12.19. The initial linear and angular velocity vec-

    tors are (15, 0, 0) m/s and (0, 0, 0) rad/s for the

    nominal case. The initial Euler angle is (0, 0, 0.9)

    rad. Safe margin is chosen as 5 m. The PID con-

    troller gains and motion planner parameters for

    potential field force are also tuned and set to fixed

    values for the verification process.

    In the simulation, the initial departure point is

    (0, 0, 20) m, and the spherical obstacle is located

    at (250, 250, 10) m with a radius r0 of 20 m.

    Therefore, the safety radius is 25 m including safe

    margin. The simulation result at the nominal para-

    meters is shown in Fig. 3.The minimum distance

    to the obstacle is obtained as 29.6166 m which is

    greater than obstacle safety radius 25 m(dmin>rn).

    0

    100

    200

    300

    400

    0

    100

    200

    300

    400

    0

    20

    40

    x (m)

    Obstacle avoidance at nominal parameters

    y (m)

    z(m)

    Start

    waypoint

    Fig. 3 Simulation result for UAV collision avoidance atnominal parameters

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    J Intell Robot Syst (2012) 65:219231 225

    This concludes that the obstacle avoidance algo-

    rithm works correctly at the nominal parameters.

    We now verify the proposed obstacle avoidance

    algorithms using the proposed optimization based

    clearance approach in 3D environment.

    5 Initial Robustness Analysis and Local

    Optimization Method

    Initial robustness analysis of the proposed algo-

    rithm is carried out in this section. Uncertainties

    are considered in the dynamic model (mass and

    two aerodynamic coefficients), and each uncertain

    parameter is allowed to vary within (10or 20)

    % of its nominal value. These are firstly con-

    sidered within lower and upper bounds, i.e. m=[1.52, 2.28] kg,Cme = [1.243, 1.017] andCt=

    [10.971, 13.409]. For the purpose of comparison,

    the uncertain parameters are normalized to have

    a variation within the range. Figure4shows varia-

    tions of the minimum distance to the obstacle with

    respect to the normalized uncertain parameters of

    mass,Cmeand Ct. There is a significant variation

    in the distance with the variations of these uncer-

    tain parameters. The minimum distance to the ob-

    stacle monotonically decreases with the increase

    of the m and Cme, while dmin increases with theincrease ofCt.

    0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.2528.5

    29

    29.5

    30

    30.5

    31

    31.5

    32

    Normalized uncertain parameters

    dmin

    (m)

    m

    Cmde

    Cdt

    Fig. 4 Mass, Cmeand Ctvariations

    5.1 Optimization-Based Worst-Case Analysis

    In this paper, the optimization clearance process is

    applied to the UAV obstacle avoidance systems.

    If the minimum distance to the obstacle is greater

    than a safety radius of an obstacle (dmin > rn)dur-

    ing the UAV moving, the proposed anti-collisionalgorithm is safe. When the optimization clear-

    ance process is applied to the system, this anti-

    collision condition is checked for all possible vari-

    ations. The local and global optimization methods

    are applied to the problem of finding a worst-case

    combination of the condition and parameters for

    the UAV collision avoidance systems. Uncertain

    parameters are considered that lies between given

    upper and lower bounds. It shall be highlighted

    that the collision avoidance algorithm is devel-

    oped only based on the kinematic model, but it hasto be implemented through aircraft dynamics as

    described to drive the aircraft to follow the desired

    total speed and attitude. There is a mismatching in

    the model structure and complexity. Furthermore,

    the parameters in the inner-loop controllers also

    have a significant influence in collision avoidance.

    This problem caused by the structural uncertainty

    is unlikely solved by formal methods.

    Simulation based optimization is proposed in

    this paper. The objective function is defined as

    dmin = min (d (t)) for t T(sec)

    s.t PL P PU(15)

    where Pis the uncertain parameters set. PL and

    PU are the lower and upper bounds of P, d(t)

    is the distance to the obstacle, T is the collision

    avoidance maneuver during the period anddminis

    the minimum distance to the obstacle.

    5.2 Local Optimization-Based

    Worst-Case Analysis

    Sequential Quadratic Programming (SQP)

    method is a standard general purpose algorithm

    for solving smooth and well-scaled nonlinear

    optimization problems when functions and

    gradients can be evaluated with high precision.

    It is an iterative method starting from an initial

    point and converging to a local minimum. The

    functionfminconis a MATLAB implementation.

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    Table 1 Local optimization results

    Algorithm Starting point Convergent point dmin(m)

    fmincon [1.52, 1.13, 11.8243] [2.28, 1.243, 10.971] 29.0235

    fmincon [1.71, 1.13, 13.409] [2.28, 1.0193, 10.971] 27.4967

    The optimization processing of fmincon consistsof three main stages:

    (i) updating of the Hessian matrix of the

    Lagrangian function, (ii) quadratic programming

    problem solution, and (iii) line search and merit

    function calculation. This iteration is repeated

    until an optimal or feasible solution is found

    Mathworks.co.uk.The local optimization method

    is applied with different starting points to the

    problem of evaluating a clearance criterion for the

    UAV obstacle avoidance systems.

    This iteration is repeated until a specified ter-mination criterion (either maximum number of

    function evaluations or convergence accuracy) is

    met. The results of the minimum distance to the

    obstacle and worst case parameters with different

    starting points are given in Table 1. The results

    clearly show thatfmincondoes not give the same

    solutions for this problem because the solution

    for a local optimization algorithm depends on the

    starting point. It does not give the true worst

    case. Therefore, global optimization methods are

    applied to find the true worst-case.

    6 Stochastic Global Optimization-Based Worst

    Case Analysis

    6.1 Genetic Algorithms

    Genetic Algorithms (GAs) are general purpose

    stochastic search and optimization algorithms,

    based on genetic and evolutionary principles. The

    theory and practice of the GA was originally in-

    vented by John Holland in 1960s and was fullyelaborated in his book Adaption in Natural and

    Artif icial Systemspublished in 1975 [22]. The ba-

    sic idea of the approach is to start with a set of

    designs, randomly generated using the allowablevalues for each design variable. Each design is also

    assigned a fitness value. The process is continued

    until a stopping criterion is satisfied or the number

    of iterations exceeds as a specified limit. Three

    genetic operators are used to accomplish this task:

    Selection, Crossover, and Mutation. Selection is

    an operator where an old design is copied into the

    new population according to the designs fitness.

    There are many different strategies to implement

    this selection operator including roulette wheel

    selection, tournament selection and stochastic uni-versal sampling. The crossover operator corre-

    sponds to allowing selected members of the new

    population to exchange characteristics of their

    designs among themselves. Crossover entails se-

    lection of starting and ending positions on a pair

    of randomly selected strings, and simply exchang-

    ing the string of 0s and 1s between these posi-

    tions. Mutation is the third step that safeguards

    the process from a complete premature loss of

    valuable genetic material during selection and

    crossover. The foregoing three steps are repeatedfor successive generations of the population until

    no further improvement in fitness is attainable

    [2325].

    GA can be applied to the UAV collision avoid-

    ance system to find the global minimum. The

    uncertain parameter set is considered here as the

    genetic representation, i.e. the chromosome. Each

    of the uncertainties corresponds to one gene. A

    binary coded string is generated to represent the

    chromosome, where each of the uncertain para-

    meters lies between the lower and upper bounds.The selection function of roulette wheel is used

    for this study. The population size and crossover

    fraction are selected as default value of 20 and

    Table 2 GA results for a UAV obstacle avoidance system

    Algorithm Starting point m(kg) Cme Ct dmin(m) Time

    GA [1.52,1.13, 11.82] 2.2773 1.0283 10.9734 27.5018 12 s

    GA [1.71,1.13, 13.409] 2.28 1.0177 10.9905 27.5032 12 s

    http://mathworks.co.uk/http://mathworks.co.uk/http://mathworks.co.uk/
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    0 20 40 60 80 10027.5

    28

    28.5

    29

    29.5

    30

    30.5

    Generation

    Fitn

    essvalue

    Best: 27.5018 Mean: 27.5057

    Best fitness

    Mean fitness

    Fig. 5 No of generations vs. fitness value

    0.8 respectively. The optimization is terminated

    after 51 iterations because the convergence accu-

    racy is met. The GA results with different start-ing points are given in Table 2. Figure 5 shows

    the number of generations versus the best fitness

    and the mean fitness values at starting points

    [1.52, 1.13, 11.82].

    6.2 GLOBAL Algorithm

    The multistart clustering algorithm presented

    in this work is based on GLOBAL developed

    by Csendes in 1988, which is a modified ver-

    sion of the stochastic algorithm by Boender

    et al (1982) implemented in FORTRAN. The

    GLOBAL method has two phases i.e. a global and

    a local one. The global phase consists of sampling

    and clustering, while the local phase is based on

    local searches. A general clustering method starts

    with the generation of a uniform sample in the

    search space (the region defined by lower and

    upper bounds). After transforming the sample

    (by selecting a user set percentage of the sample

    points with the lowest function values), the clus-

    tering procedure is applied. Then, the local search

    is started from those points which have not been

    assigned to a cluster. GLOBAL uses the Single

    Linkage clustering rule [26].

    The new implementation GLOBALm, which

    has been written in MATLAB, is freely available

    for academic purposes. It is the bound constrained

    global optimization problems with a black-box

    type objective function. GLOBALm has differ-

    ent local optimization methods which are capable

    of handling constraints. The UNIRANDI localsearch method is part of GLOBAL package while

    the BFGS (Broyden-Fletcher-Goldfarb-Shanno)

    local search is part of the MATLAB package.

    GLOBAL has six parameters to set: the number

    of sample points, the number of best points se-

    lected, the stopping criterion parameter for local

    search, the maximum number of function eval-

    uations for local search, the maximum number

    of local minima to explore, and the used lo-

    cal method. All these parameters have a default

    value [26].The GLOBAL optimization with UNIRANDI

    local search method is applied to find the global

    solution for the UAV obstacle avoidance sys-

    tem. The results with different numbers of the

    sampling points are given in Table3. GLOBAL

    algorithm gives the nearly same solutions with

    different number of sampling points. It takes 1,112

    functions evaluation with 200 sampling points

    while 9,810 functions evaluations with 500 sam-

    pling points. Thirty six local minimum are found

    at 500 sampling points while eight found at 200sampling points. GA and GLOBAL algorithms

    are performed well for this case study. However,

    both these algorithms cannot guarantee the worst

    case is found.

    7 Deterministic Global Optimization-Based

    Worst Case Analysis

    7.1 DIRECT Method

    The disadvantage of the stochastic global opti-

    mization methods including GA and GLOBAL

    Table 3 GLOBAL results for a UAV obstacle avoidance system

    No of sample m (kg) Cme Ct dmin(m) Fun.Evalu taken Time

    200 2.2063 1.0265 11.2221 27.748 1,112 13 min

    500 2.1798 1.0171 10.971 27.494 9,810 2 h 15 min

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    228 J Intell Robot Syst (2012) 65:219231

    algorithms is that there are no formal proofs of

    convergence. In order to avoid this problem, a de-

    terministic global optimization algorithm known

    as DIRECT method (DIviding RECTangles) is

    also considered in the verification process for

    the obstacle avoidance. The DIRECT algorithm

    was developed by Jones et al in 1993 [27], whichguarantees to the convergence to the globally

    optimal if the objective function is continuous

    or at least continuous in the neighborhood of

    the global optimum. The global convergence may

    come at the expense of a large and exhaustive

    search over the domain. The DIRECT algorithm

    was created in order to solve difficult global opti-

    mization problems with bound constraints and a

    real-valued objective function. The DIRECT

    method does not require any derivative informa-

    tion. It is a modification of the standard Lip-schitzian optimization method. This global search

    algorithm can be very useful when the objective

    function is a black-box function. The DIRECT

    algorithm is described below[23,28,29].

    7.1.1 Normalization and Division

    of the Hyper-cube

    DIRECT begins the optimization by transforming

    the domain of the problem into a unit hyper-cube.That is,

    =X RN : 0 Xi 1

    (16)

    The algorithm works in this normalized space.

    Let c1 be the center point of this hypercube and

    evaluate f(c1). The next step is to divide this

    hyper-cube by evaluating the function values at

    the points c1 ei, i= 1, 2, . . .N, where is one-

    third of the side length of the hyper-cube, and eiisthe i th unit vector. That is, a hyper-cube is divided

    into three hyper-rectangles in each dimension.

    The DIRECT algorithm chooses to leave the

    best function values in the largest space; therefore,

    the smallestican be defined as

    i = min (f(c1+ ei) , f(c1 ei)) , 1 i N

    (17)

    and then divide the dimension with the smallest

    i into thirds, so that c1 ei, i= 1, 2, . . .N are

    the centers of the new hyper-rectangles. This pat-

    tern is repeated for all dimensions on the centre

    hyper-rectangle, choosing the next dimension by

    determining the next smallesti.

    7.1.2 Potentially Optimal Hyper-rectangles

    DIRECT then determines which rectangles are

    potentially optimal, and should be divided in this

    iteration.

    Let> 0be a positive constant and let fminbe

    the current best function value. A hyper-rectanglejis potentially optimal if there exists some K> 0

    such that

    f

    cj

    Kdj f(ci) Kdi, iandf

    cj

    Kdj fmin | fmin|(18)

    In (18), cj is the center point of the hyper-

    rectangle j, and dj defines a measure for the

    hyper-rectangles. Jones et al. chose to use the dis-

    tance from center point cj to its vertices as the

    measure and also concluded that a good value for

    is 1 104. Figure6illustrates this definition.

    7.1.3 Division of the Hyper-rectangles

    Once a hyper-rectangle has been identified

    as potentially containing the optimal solution,

    Fig. 6 Hyper-rectangles on the piecewise linear curve arepotentially optimal[23]

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    Table 4 DIRECT results for a UAV obstacle avoidance system

    Algorithm Iteration m (kg) Cme Ct dmin(m) Fun.Evalu taken Time

    Direct 500 2.28 1.017 10.976 27.498 18,505 3 h 35 m

    DIRECT divides this hyper-rectangle into smaller

    hyper-rectangles. DIRECT divides the hyper-rectangles by performing division only in the

    dimensions with the longest side length.

    The sequence of the dimensions to be divided

    is determined byjwhich is defined as

    j= min

    f

    ci+ iej

    , f

    ci iej

    , j I

    (19)

    where iis one-third the length of the longest side

    of hyper-rectangle I,ejis the jth unit vector, and

    Iis the set of all dimensions of the longest side

    length. This process is repeated for all dimensions

    in I.

    7.2 Simulation Results

    The DIRECT algorithm is applied to the UAV

    obstacle avoidance verification process, and the

    results are given in Table4.The DIRECT method

    requires no initial guesses but operates on the pa-

    rameters upper and lower bounds. The DIRECTalgorithm terminates as soon as it exceeds the

    given iterations. The history of iteration versus

    0 100 200 300 400 50027.4

    27.6

    27.8

    28

    28.2

    28.4

    28.6

    28.8

    29

    29.2

    Iteration

    fmin

    Iteration Statistics

    Fig. 7 Directalgorithm-iteration vs. fitness value

    fitness value is shown in Fig. 7. All optimization

    algorithms are performed in MATLAB 2010a andIntel (R) Core(TM) 2 Duo CPU (3.16 GHz).

    DIRECT takes 3 h 35 min to converge to the

    global minimum. Compared to the stochastic

    global algorithms, GA and GLOBAL algorithm

    are performed well for this case study, but GA

    performs faster. However, these are stochastic

    global algorithms and there is no confidence to

    establish the true worst case. The DIRECT algo-

    rithm can guarantee finding the worst case in this

    application, but the computation time is high.

    These worst-case condition and worst-case pa-rameters identified in the verification process are

    further validated with simulation response shown

    in Fig. 8. The time versus distance to the obsta-

    cle at the nominal and worst-case parameters is

    shown in Fig. 9. The worst-case minimum dis-

    tance to the obstacle dmin is 27.4982 m which

    is greater than the specified safety radius of the

    obstacle. This concludes that the obstacle avoid-

    ance algorithm and the controller provide ade-

    quate performance at the worst-case parameters.

    Furthermore, in the presence of all the describedvariations, the safety margin for anti-collision is

    respected.

    0100

    200300

    400

    0

    100

    200

    300

    400

    -10

    0

    10

    20

    30

    40

    50

    x (m)

    Obstacle avoidance at worst case parameters

    y (m)

    z(m)

    Start

    waypoint

    Fig. 8 Simulation response at worst-case parameters

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    24 26 28 30 32 3420

    25

    30

    35

    40

    45

    50

    55

    60

    Time (sec)

    Distance

    d(t),(m)

    at nominal parameters

    at worst case parameters

    Fig. 9 Time vs distance to the obstacle at nominal andworst case parameters

    8 Conclusions

    In this paper, optimization based clearance

    process of obstacle avoidance systems is applied

    to verify collision avoidance algorithms for UAVs.

    The key idea in this verification approach is that it

    is not necessary for an optimization algorithm to

    evaluate a cost function over all possible solutions

    in order to find the optimal solution. However

    different from many optimization problems, it isimportant to find all the possible worst cases in

    order to verify safety critical functionalities like

    obstacle avoidance. This requires an optimization

    algorithm that converges to the global optimal

    solution.

    In developing optimization based worst case

    analysis for verification of collision avoidance al-

    gorithms, the minimum distance to the obstacle

    during collision avoidance maneuver is defined as

    the cost function. To demonstrate the concept,

    a 6DOF UAV model is used in the case study

    with a designed collision avoidance algorithm, and

    mass and two aerodynamic coefficients variations

    are considered for the verification purpose. The

    local optimization method does not give a unique

    solution as different worst cases are identified

    when the optimization starts from different initial

    conditions. Therefore, the local optimization is

    not suitable for verification of collision avoidance

    algorithms for this case study.

    To overcome this problem, global optimiza-

    tion algorithms are studied. Stochastic global opti-

    mization algorithms including GA and GLOBAL

    methods have been applied to the problem. GA

    and GLOBAL algorithm perform well for this

    UAV problem. The deterministic global optimiza-

    tion of the DIRECT method has also been investi-gated for the worst-case analysis. Compared with

    other global optimization algorithms in this study,

    the DIRECT algorithm can guarantee finding the

    worst case, although it takes more time to con-

    verge. It shall be highlighted that in developing

    collision avoidance algorithms using the potential

    field method, the UAV is considered as a mass

    point and only the kinematic model is used. How-

    ever in real implementation of collision avoidance

    maneuver, the UAV has to be controlled to follow

    the desirable total velocity and attitude. There-fore, the UAV dynamics and the influence of the

    inner loop controllers for tracking reference speed

    and attitude provided by the collision avoidance

    algorithm must be taken into account in order

    to fully understand the behavior of the collision

    avoidance algorithm. This is particularly impor-

    tant for very close maneuvers like collision avoid-

    ance. The work presented in this paper provides

    a framework of taking into account the different

    levels of the model complexity used in the differ-

    ent stages of autonomous control development.It can significantly improve the efficiency of the

    verification process by automatically searching

    the worst cases without the need to exhaustively

    evaluate all possible combinations of variations.

    Further work will be on applying the proposed

    verification approach to an environment with dy-

    namic obstacles for UAVs.

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